# Finding the median

• Jul 25th 2012, 10:11 AM
dannyc
Finding the median
The sum of 5 consecutive integers is 10n + 5. What is the median of the 5 integers in terms of n?

How can you prove that it must be 2n + 1?
• Jul 25th 2012, 11:41 AM
HallsofIvy
Re: Finding the median
Call the first of the consecutive integers "i". Then the next four integers are i+1, i+2, i+3, and i+4. The sum of those five numbers is i+ (i+1)+ (i+ 3)+ (i+ 4)+ (i+ 5)= 5i+ 10= 10n+ 5. Solve that for i in terms of n. The "median" is the middle number, i+ 2.
• Jul 25th 2012, 06:14 PM
dannyc
Re: Finding the median
How does 5i+10 imply 10n+5 ? I don't see it nor do I see how i + 2 implies 2n + 1.
• Jul 25th 2012, 09:58 PM
earboth
Re: Finding the median
Quote:

Originally Posted by HallsofIvy
Call the first of the consecutive integers "i". Then the next four integers are i+1, i+2, i+3, and i+4. The sum of those five numbers is i+ (i+1)+ (i+ 3)+ (i+ 4)+ (i+ 2)= 5i+ 10= 10n+ 5. Solve that for i in terms of n. The "median" is the middle number, i+ 2.

Quote:

Originally Posted by dannyc
The sum of 5 consecutive integers is 10n + 5. <--- your own words. Remember?
What is the median of the 5 integers in terms of n?

How can you prove that it must be 2n + 1?

Quote:

Originally Posted by dannyc
How does 5i+10 imply 10n+5 ? I don't see it nor do I see how i + 2 implies 2n + 1.

You are supposed to know what a median means or is.

In your case the median is the 3rd term, that means (i + 2). Since i = 2n - 1 the term i + 2 = 2n + 1.
• Jul 26th 2012, 04:07 AM
dannyc
Re: Finding the median
Right, but I still do not know how we can set i + 2 = 2n + 1. Why can that conclusion be made algebraically?
• Jul 26th 2012, 04:55 AM
earboth
Re: Finding the median
Quote:

Originally Posted by dannyc
Right, but I still do not know how we can set i + 2 = 2n + 1. Why can that conclusion be made algebraically?

1.

$\displaystyle \underbrace{i+(i+1)+\overbrace{(i+2)}^{median}+(i+ 3)+(i+4)}_{\text{according to HallsofIvy}} = \overbrace{10n+5}^{\text{according to dannyc}}$

$\displaystyle 5i+10=10n+5$

$\displaystyle 5i=10n-5=5(2n-1)$

$\displaystyle \boxed{i=2n-1}$ Now add 2 on both sides of this equation:

$\displaystyle i+\color{red}2 \color{black}= (2n-1)+\color{red}2$

$\displaystyle \boxed{i+2 = 2n+1}$
• Jul 26th 2012, 05:17 AM
HallsofIvy
Re: Finding the median
Quote:

Originally Posted by dannyc
How does 5i+10 imply 10n+5 ? I don't see it nor do I see how i + 2 implies 2n + 1.

I don't know what you mean by "implies" here. I said they were equal and that is true because YOU said it was:
"The sum of 5 consecutive integers is 10n + 5".